First derivative of the legendre polynomials

[kilojoules]

show that the first derivative of the legendre polynomials satisfy a self-adjoint differential equation with eigenvalue λ=n(n+1)-2


The attempt at a solution:

(1-x^2 ) P_n^''-2xP_n^'=λP_n
λ = n(n + 1) - 2 and (1-x^2 ) P_n^''-2xP_n^'=nP_(n-1)^'-nP_n-nxP_n^'
∴nP_(n-1)^'-nP_n-nxP_n^'=( n(n + 1) - 2 )P_n
For n=2
2P_1^'-2P_2-2xP_2^'=4P_n
Using legendre polynomial properties we have
-9x^2+2x+1=2(3x^2-1)
∴-15x^2+2x+3=0
I don’t know what to do from here.

 

[HallsofIvy]

Did you consider just replacing the first derivative with just "y" so that the equation becomes
$$(1- x^2)y'- 2xy= \lambda P_n$$
Differentiating
$$(1- x^2)y''- 4xy'- 2y= \lambda P_n'= \lambda y$$

$$(1- x^2)y''- 4xy'= (2+ \lambda)y$$

 

[kilojoules]

Thanks Hallsofvy